Optimal Sliding and Decoupled Sliding Mode Tracking Control by Multi-objective Particle Swarm Optimization and Genetic Algorithms

Abstract

The objective of this chapter is to present an optimal robust control approach based upon smart multi-objective optimization algorithms for systems with challenging dynamic equations in order to minimize the control inputs and tracking and position error. To this end, an optimal sliding and decoupled sliding mode control technique based on three multi-objective optimization algorithms, that is, multi-objective periodic CDPSO, modified NSGAII and Sigma method is presented to control two dynamic systems including biped robots and ball and beam systems. The control of biped robots is one of the most challenging topics in the field of robotics because the stability of the biped robots is usually provided laboriously regarding the heavily nonlinear dynamic equations of them. On the other hand, the ball and beam system is one of the most popular laboratory models used widely to challenge the control techniques. Sliding mode control (SMC) is a nonlinear controller with characteristics of robustness and invariance to model parametric uncertainties and nonlinearity in the dynamic equations. Hence, optimal sliding mode tracking control tuned by multi-objective optimization algorithms is utilized in this study to present a controller having exclusive qualities, such as robust performance and optimal control inputs. To design an optimal control approach, multi-objective particle swarm optimization (PSO) called multi-objective periodic CDPSO introduced by authors in their previous research and two notable smart multi-objective optimization algorithms, i.e. modified NSGAII and the Sigma method are employed to ascertain the optimal parameters of the control approach with regard to the design criteria. In comparison, genetic algorithm optimization operates based upon reproduction, crossover and mutation; however particle swarm optimization functions by means of a convergence and divergence operator, a periodic leader selection method, and an adaptive elimination technique. When the multi-objective optimization algorithms are applied to the design of the controller, there is a trade-off between the tracking error and control inputs. By means of optimal points of the Pareto front obtained from the multi-objective optimization algorithms, plenty of opportunity is provided to engineers to design the control approach. Contrasting the Pareto front obtained by multi-objective periodic CDPSO with two noteworthy multi-objective optimization algorithms i.e. modified NSGAII and Sigma method dramatizes the excellent performance of multi-objective periodic CDPSO in the design of the control method. Finally, the optimal sliding mode tracking control tuned by CDPSO is applied to the control of a biped robot walking in the lateral plane on slope and the ball and beam system. The results and analysis prove the efficiency of the control approach with regard to providing optimal control inputs and low tracking and position errors.

Appendix

The equivalent sliding mode control inputs for the biped robot are as follows

$$\begin{aligned} {\text {u}}_{{{\text {eq}}1}}&= - 0.2260855175\,{{\dot{\uptheta }}}_{2} - 5.376931543\,{{\uptheta }}_{3} + 0.5609845516{\text {t}}^{3} - 24.77733929{\text {t}} \\&\quad + 11.04563973 + 0.3042022263{{\dot{\uptheta }}}_{3} - 52.53069026{{\uptheta }}_{1} \\&\quad - 23.33466928{{\lambda }}_{2} {{\dot{\uptheta }}}_{2} - 2.116394625{{\lambda }}_{3} {{\dot{\uptheta }}}_{3} - 61.09883670{{\lambda }}_{1} {{\dot{\uptheta }}}_{1} \\&\quad + 1.093385806{{\dot{\uptheta }}}_{1} - 2.335441469{{\lambda }}_{3} - 5.878003190{{\lambda }}_{2} \\&\quad - 16.56389463{{\lambda }}_{1} - 14.61433450{{\uptheta }}_{2} - 1.029058915{{\lambda }}_{2} {\text {t}}^{2} \\&\quad + 5.282969124{{~~\lambda }}_{2} {\text {t}} + 11.12677748{\text {t}}^{2} - 0.3123798467{{\lambda }}_{3} {\text {t}}^{2} \\&\quad + 1.744755729{{\lambda }}_{3} {\text {t}} - 3.024392417{{\lambda }}_{1} {\text {t}}^{2} + 15.22583011{{\lambda }}_{1} {\text {t}} \end{aligned}$$

$$\begin{aligned} {\text {u}}_{{{\text {eq}}2}}&= 2.127074613 \left( {10^{{ - 10}} } \right) {{\dot{\uptheta }}}_{2} + 3.014817654{{\uptheta }}_{3} - 0.1832721690{\text {t}}^{3} \\&\quad - 7.435154298{\text {t}} + 0.06684314392{{\dot{\uptheta }}}_{3} + 0.6081082555{{\uptheta }}_{1} \\&\quad - 12.97075008{{\lambda }}_{2} {{\dot{\uptheta }}}_{2} - 2.116394625{{\lambda }}_{3} {{\dot{\uptheta }}}_{3} - 6.759058956{{\lambda }}_{1} {{\dot{\uptheta }}}_{1} \\&\quad + 1.093385810{{\dot{\uptheta }}}_{1} - 2.335441469{{\lambda }}_{3} - 3.267331946{{\lambda }}_{2} \\&\quad - 1.832380883{{\lambda }}_{1} - 14.19340124{{\uptheta }}_{2} - 0.5720100787{{\lambda }}_{2} {\text {t}}^{2} \\&\quad + 2.936577819{{\lambda }}_{2} {\text {t}} + 3.182845520{\text {t}}^{2} + 9.317127676 \\&\quad - 0.3123798467{{\lambda }}_{3} {\text {t}}^{2} + 1.744755729{{\lambda }}_{3} {\text {t}} - 0.3345734183{{\lambda }}_{1} {\text {t}}^{2} \\&\quad + 1.684357492{{\lambda }}_{1} {\text {t}} \end{aligned}$$

$$\begin{aligned} {\text {u}}_{{{\text {eq}}3}}&= - 3.400956231\left( {10^{{ - 11}} } \right) {{\dot{\uptheta }}}_{2} + 3.029347274{{\uptheta }}_{3} - 0.2748341166{\text {t}}^{3} \\&\quad - 3.658549547{\text {t}} + 11.95619521 + 1.750512761\left( {10^{{ - 10}} } \right) {{\dot{\uptheta }}}_{3} \\&\quad + 0.1293520741{{\uptheta }}_{1} - 2.497500599{{\lambda }}_{2} {{\dot{\uptheta }}}_{2} - 3.630907800{{\lambda }}_{3} {{\dot{\uptheta }}}_{3} \\&\quad + 3.604860241{{\lambda }}_{1} {{\dot{\uptheta }}}_{1} + 0.2233967865\dot{{{\uptheta }}}_{1} - 4.006706757{{~~\lambda }}_{3} \\&\quad - 0.6291204009{{~\lambda }}_{2} + 0.9772776114{{\lambda }}_{1} + 1.33021167{\text {t}}^{2} \\&\quad - 0.01647731197{{\uptheta }}_{2} - 0.1101397764{{~~\lambda }}_{2} {\text {t}}^{2} + 0.5654341356{{\lambda }}_{2} {\text {t}} \\&\quad - 0.8983311722{{~\lambda }}_{1} {\text {t}} - 0.5359219913{{~\lambda }}_{3} {\text {t}}^{2} + 2.993320390{{~\lambda }}_{3} {\text {t}} \\&\quad + 0.1784405819{{\lambda }}_{1} {\text {t}}^{2} \end{aligned}$$